Interest is a fee A fee is the price one pays as remuneration for services. Fees usually allow for overhead, wages, costs, and markup paid on borrowed assets. It is the price paid for the use of borrowed money,^[1] or, money earned by deposited funds.^[2] Assets In financial accounting, assets are economic resources. Anything tangible or intangible that is capable of being owned or controlled to produce value and that is held to have positive economic value is considered an asset. Simplistically stated, assets represent ownership of value that can be converted into cash . The balance sheet of a firm that are sometimes lent with interest include money Money is any object that is generally accepted as payment for goods and services and repayment of debts in a given country or socio-economic context. The main functions of money are distinguished as: a medium of exchange; a unit of account; a store of value; and, occasionally, a standard of deferred payment, shares In financial markets, a share is a unit of account for various financial instruments including stocks , and investments in limited partnerships, and REITs. The common feature of all these is equity participation (limited in the case of preference shares), consumer goods In economics final goods are goods that are ultimately consumed rather than used in the production of another good. For example, a car sold to a consumer is a final good; the components such as tires sold to the car manufacturer are not; they are intermediate goods used to make the final good through hire purchase Hire purchase is the legal term for a contract, in this persons usually agree to pay for goods in parts or a percentage at a time. It was developed in the United Kingdom and can now found in China, Japan, Malaysia, India, Australia, and New Zealand. It is also called closed-end leasing. In cases where a buyer cannot afford to pay the asked price, major assets such as aircraft Aircraft finance refers to financing for the purchase and operation of aircraft. Complex aircraft finance shares many characteristics with maritime finance, and to a lesser extent with project finance, and even entire factories in finance lease The U.S. Financial Accounting Standards Board and the International Accounting Standards Board announced in 2006 a joint project to comprehensively review lease accounting standards. In July 2008, the boards decided to defer any changes to lessor accounting, while continuing with the projects for lessee accounting, with the stated intention to arrangements. The interest is calculated upon the value of the assets in the same manner as upon money. Interest can be thought of as "rent Renting is an agreement where a payment is made for the temporary use of a good, service or property owned by another. A gross lease is when the tenant pays a flat rental amount and the landlord pays for all property charges regularly incurred by the ownership from lawnmowers and washing machines to handbags and jewellry of money". When money is deposited in a bank, interest is typically paid to the depositor as a percentage of the amount deposited; when money is borrowed, interest is typically paid to the lender as a percentage of the amount owed. The percentage of the principal that is paid as a fee over a certain period of time (typically one month or year), is called the interest rate An interest rate is the rate at which interest is paid by a borrower for the use of money that they borrow from a lender. For example, a small company borrows capital from a bank to buy new assets for their business, and in return the lender receives interest at a predetermined interest rate for deferring the use of funds and instead lending it to.

Interest is compensation to the lender, and for forgoing other useful investments Investment is the commitment of money or capital to purchase financial instruments or other assets in order to gain profitable returns in form of interest, income, or appreciation of the value of the instrument. It is related to saving or deferring consumption. Investment is involved in many areas of the economy, such as business management and that could have been made with the loaned asset. These forgone investments are known as the opportunity cost Opportunity cost is the cost related to the next-best choice available to someone who has picked between several mutually exclusive choices. It is a key concept in economics. It has been described as expressing "the basic relationship between scarcity and choice." The notion of opportunity cost plays a crucial part in ensuring that. Instead of the lender using the assets directly, they are advanced to the borrower. The borrower then enjoys the benefit of using the assets ahead of the effort required to obtain them, while the lender enjoys the benefit of the fee paid by the borrower for the privilege. Interest also compensates the lender for the risk of losing the principal, called credit risk Credit risk is an investor's risk of loss arising from a borrower who does not make payments as promised. Such an event is called a default. Another term for credit risk is default risk. In economics, interest is considered the price of credit.

History of interest

In ancient biblical Israel, it was against the Law of Moses to charge interest on private loans^[3].During the Middle Ages The Middle Ages is a period of European history from the 5th century to the 15th century. The period followed the fall of the Western Roman Empire in 476, and preceded the Early Modern Era. It is the middle period in a three-period division of history: Classical, Medieval, and Modern. The term "Middle Ages" (medium aevum) was coined in, time was considered to be property of God God is the English name given to the singular omnipotent being in theistic and deistic religions who is either the sole deity in monotheism, or a single deity in polytheism. Therefore, to charge interest was considered to be commerce with God's property.^{[citation needed]} Also, St. Thomas Aquinas Saint Thomas Aquinas, O.P. was an Italian priest of the Catholic Church in the Dominican Order, and an immensely influential philosopher and theologian in the tradition of scholasticism, known as Doctor Angelicus and Doctor Communis. He is frequently referred to as Thomas because "Aquinas" refers to his residence rather than his surname, the leading theologian of the Catholic Church The Catholic Church, also known as the Roman Catholic Church, is the world's largest Christian church, with more than a billion members. The Church's leader is the Pope who holds supreme authority in concert with the College of Bishops of which he is the head. A communion of the Western church and 22 autonomous Eastern Catholic churches (called, argued that the charging of interest is wrong because it amounts to "double charging Usury originally meant the charging of interest on loans. This included charging a fee for the use of money, such as at a bureau de change. After interest became acceptable, usury came to mean the interest above the rate allowed by law. In common usage today, the word means the charging of unreasonable or relatively high rates of interest. The", charging for both the thing and the use of the thing. The church regarded this as a sin of usury Usury originally meant the charging of interest on loans. This included charging a fee for the use of money, such as at a bureau de change. After interest became acceptable, usury came to mean the interest above the rate allowed by law. In common usage today, the word means the charging of unreasonable or relatively high rates of interest. The; nevertheless, this rule was never strictly obeyed and eroded gradually until it disappeared during the industrial revolution.^{[citation needed]}

Usury Usury originally meant the charging of interest on loans. This included charging a fee for the use of money, such as at a bureau de change. After interest became acceptable, usury came to mean the interest above the rate allowed by law. In common usage today, the word means the charging of unreasonable or relatively high rates of interest. The has always been viewed negatively by the Roman Catholic Church. The Second Lateran Council The Second Council of the Lateran is believed to have been the Tenth Ecumenical Council by Roman Catholics. It was held by Pope Innocent II in April 1139, and was attended by close to a thousand clerics. Its immediate task was to neutralise the after-effects of the schism, which had arisen after the death of Pope Honorius II in February 1130 and condemned any repayment of a debt with more money than was originally loaned, the Council of Vienna explicitly prohibited usury and declared any legislation tolerant of usury to be heretical, and the first scholastics reproved the charging of interest. In the medieval economy, loans were entirely a consequence of necessity (bad harvests, fire in a workplace) and, under those conditions, it was considered morally reproachable to charge interest.^{[citation needed]} It was also considered morally dubious, since no goods were produced through the lending of money, and thus it should not be compensated, unlike other activities with direct physical output such as blacksmithing or farming.^[4]

As Jewish citizens were ostracized from most professions by local rulers, the church and the guilds, they were pushed into marginal occupations considered socially inferior, such as tax and rent collecting and moneylending. Natural tensions between creditors and debtors were added to social, political, religious, and economic strains. ...financial oppression of Jews tended to occur in areas where they were most disliked, and if Jews reacted by concentrating on moneylending to non-Jews, the unpopularity — and so, of course, the pressure — would increase. Thus the Jews became an element in a vicious circle. The Christians, on the basis of the Biblical rulings, condemned interest-taking absolutely, and from 1179 those who practiced it were excommunicated. Catholic autocrats frequently imposed the harshest financial burdens on the Jews. The Jews reacted by engaging in the one business where Christian laws actually discriminated in their favor, and became identified with the hated trade of moneylending.

Interest has often been looked down upon in Islamic civilization as well for the same reason for which usury was forbidden by the Catholic Church, with most scholars agreeing that the Qur'an explicitly forbids charging interest. Medieval jurists therefore developed several financial instruments to encourage responsible lending.

In the Renaissance The Renaissance was a cultural movement that spanned roughly the 14th to the 17th century, beginning in Florence in the Late Middle Ages and later spreading to the rest of Europe. The term is also used more loosely to refer to the historic era, but since the changes of the Renaissance were not uniform across Europe, this is a general use of the era, greater mobility of people facilitated an increase in commerce and the appearance of appropriate conditions for entrepreneurs An entrepreneur is a person who has possession of a new enterprise, venture or idea and assumes significant accountability for the inherent risks and the outcome.[note 1] The term is originally a loanword from French and was first defined by the Irish economist Richard Cantillon. Entrepreneur in English is a term applied to the type of personality to start new, lucrative businesses. Given that borrowed money was no longer strictly for consumption but for production as well, interest was no longer viewed in the same manner. The School of Salamanca The School of Salamanca is the renaissance of thought in diverse intellectual areas by Spanish theologians, rooted in the intellectual and pedagogical work of Francisco de Vitoria. From the beginning of the 16th century the traditional Catholic conception of man and of his relation to God and to the world had been assaulted by the rise of humanism, elaborated on various reasons that justified the charging of interest: the person who received a loan benefited, and one could consider interest as a premium paid for the risk taken by the loaning party. There was also the question of opportunity cost Opportunity cost is the cost related to the next-best choice available to someone who has picked between several mutually exclusive choices. It is a key concept in economics. It has been described as expressing "the basic relationship between scarcity and choice." The notion of opportunity cost plays a crucial part in ensuring that, in that the loaning party lost other possibilities of using the loaned money. Finally and perhaps most originally was the consideration of money itself as merchandise, and the use of one's money as something for which one should receive a benefit in the form of interest. Martín de Azpilcueta Martín de Azpilcueta , or Doctor Navarrus, was an important Spanish canonist and theologian in his time also considered the effect of time. Other things being equal, one would prefer to receive a given good now rather than in the future. This preference There is no absolute distinction that separates "high" and "low" time preference, only comparisons with others either individually or in aggregate. Someone with a high time preference is focused substantially on his well-being in the present and the immediate future relative to the average person, while someone with low time indicates greater value. Interest, under this theory, is the payment for the time the loaning individual is deprived of the money.

Economically, the interest rate An interest rate is the rate at which interest is paid by a borrower for the use of money that they borrow from a lender. For example, a small company borrows capital from a bank to buy new assets for their business, and in return the lender receives interest at a predetermined interest rate for deferring the use of funds and instead lending it to is the cost of capital and is subject to the laws of supply and demand Supply and demand is an economic model of price determination in a market. It concludes that in a competitive market, price will function to equalize the quantity demanded by consumers, and the quantity supplied by producers, resulting in an economic equilibrium of price and quantity of the money supply In economics, the money supply or money stock, is the total amount of money available in an economy at a particular point in time. There are several ways to define "money," but standard measures usually include currency in circulation and demand deposits. The first attempt to control interest rates through manipulation of the money supply was made by the French Central Bank The Banque de France is the central bank of France; it is linked to the European Central Bank . Its main charge is to implement the interest rate policy of the European System of Central Banks (ESCB). It is headquartered in Paris in 1847 Year 1847 was a common year starting on Friday (link will display the full calendar) of the Gregorian Calendar (or a common year starting on Wednesday of the 12-day slower Julian calendar).

The first formal studies of interest rates and their impact on society were conducted by Adam Smith Adam Smith was a Scottish moral philosopher and a pioneer of political economics. One of the key figures of the Scottish Enlightenment, Smith is the author of The Theory of Moral Sentiments and An Inquiry into the Nature and Causes of the Wealth of Nations. The latter, usually abbreviated as The Wealth of Nations, is considered his magnum opus and, Jeremy Bentham Jeremy Bentham was an English jurist, philosopher, and legal and social reformer. He became a leading theorist in Anglo-American philosophy of law, and a political radical whose ideas influenced the development of welfarism. He is best known for his advocacy of utilitarianism and animal rights, and the idea of the panopticon and Mirabeau during the birth of classic economic thought.^{[citation needed]} In the early 20th century The 20th century of the Common Era began on January 1, 1901 and ended on December 31, 2000. According to the Gregorian calendar, 2000 was the first century leap year since 1600, Irving Fisher Irving Fisher was an American economist, health campaigner, and eugenicist, and one of the earliest American neoclassical economists, though he later rejected the underlying theory of general equilibrium, and his later work on debt deflation is instead considered in the Post-Keynesian school. Although he was perhaps the first celebrity economist, made a major breakthrough in the economic analysis of interest rates by distinguishing nominal interest from real interest. Several perspectives on the nature and impact of interest rates have arisen since then.

The latter half of the 20th century saw the rise of interest-free Islamic banking and finance, a movement that attempts to apply religious law developed in the medieval period to the modern economy. Some entire countries, including Iran, Sudan, and Pakistan, have taken steps to eradicate interest from their financial systems entirely.^{[citation needed]} Rather than charging interest, the interest-free lender charges a "fee" for the service of lending. As any such fee can be shown to be mathematically identical to an interest charge, the distinction between "interest-free" banking and "for-interest" banking is merely one of semantics

Types of interest

Simple interest

Simple interest is calculated only on the principal amount, or on that portion of the principal amount that remains unpaid.

The amount of simple interest is calculated according to the following formula:

where r is the period interest rate (I/m), B₀ the initial balance and m the number of time periods elapsed.

To calculate the period interest rate r, one divides the interest rate I by the number of periods m.

For example, imagine that a credit card holder has an outstanding balance of $2500 and that the simple interest rate An interest rate is the rate at which interest is paid by a borrower for the use of money that they borrow from a lender. For example, a small company borrows capital from a bank to buy new assets for their business, and in return the lender receives interest at a predetermined interest rate for deferring the use of funds and instead lending it to is 12.99% per annum. The interest added at the end of 3 months would be,

and he would have to pay $2581.19 to pay off the balance at this point.

If instead he makes interest-only payments for each of those 3 months at the period rate r, the amount of interest paid would be,

His balance at the end of 3 months would still be $2500.

In this case, the time value of money For example, 100 dollars of today's money invested for one year and earning 5 percent interest will be worth 105 dollars after one year. Therefore, 100 dollars paid now or 105 dollars paid exactly one year from now both have the same value to the recipient who assumes 5 percent interest; using time value of money terminology, 100 dollars invested is not factored in. The steady payments have an additional cost that needs to be considered when comparing loans. For example, given a $100 principal:

• Credit card debt where $1/day is charged: 1/100 = 1%/day = 7%/week = 365%/year.
• Corporate bond where the first $3 are due after six months, and the second $3 are due at the year's end: (3+3)/100 = 6%/year.
• Certificate of deposit (GIC A Guaranteed Investment Certificate or GIC is a Canadian investment that offers a guaranteed rate of return over a fixed period of time, most commonly issued by trust companies or banks. Due to its low risk profile, the return is generally less than other investments such as stocks, bonds, or mutual funds. It is similar to a Time deposit as known) where $6 is paid at the year's end: 6/100 = 6%/year.

There are two complications involved when comparing different simple interest bearing offers.

1. When rates are the same but the periods are different a direct comparison is inaccurate because of the time value of money For example, 100 dollars of today's money invested for one year and earning 5 percent interest will be worth 105 dollars after one year. Therefore, 100 dollars paid now or 105 dollars paid exactly one year from now both have the same value to the recipient who assumes 5 percent interest; using time value of money terminology, 100 dollars invested. Paying $3 every six months costs more than $6 paid at year end so, the 6% bond cannot be 'equated' to the 6% GIC.
2. When interest is due, but not paid, does it remain 'interest payable', like the bond's $3 payment after six months or, will it be added to the balance due? In the latter case it is no longer simple interest, but compound interest.

A bank account that offers only simple interest, that money can freely be withdrawn from is unlikely, since withdrawing money and immediately depositing it again would be advantageous.

Compound interest

Compound interest is very similar to simple interest; however, with time, the difference becomes considerably larger. This difference is because unpaid interest is added to the balance due. Put another way, the borrower is charged interest on previous interest. Assuming that no part of the principal or subsequent interest has been paid, the debt is calculated by the following formulas:

where I_comp is the compound interest, B₀ the initial balance, B_m the balance after m periods (where m is not necessarily an integer) and r the period rate.

For example, if the credit card holder above chose not to make any payments, the interest would accumulate

So, at the end of 3 months the credit card holder's balance would be $2582.07 and he would now have to pay $82.07 to get it down to the initial balance. Simple interest is approximately the same as compound interest over short periods of time, so frequent payments are the least expensive repayment strategy.

A problem with compound interest is that the resulting obligation can be difficult to interpret. To simplify this problem, a common convention in economics is to disclose the interest rate as though the term were one year, with annual compounding, yielding the effective interest rate The effective interest rate, effective annual interest rate, annual equivalent rate or simply effective rate is the interest rate on a loan or financial product restated from the nominal interest rate as an interest rate with annual compound interest payable in arrears. It is used to compare the annual interest between loans with different. However, interest rates in lending A loan is a type of debt. Like all debt instruments, a loan entails the redistribution of financial assets over time, between the lender and the borrower are often quoted as nominal interest rates In finance and economics nominal interest rate or nominal rate of interest refers to the rate of interest before adjustment for inflation ; or, for interest rates "as stated" without adjustment for the full effect of compounding (also referred to as the nominal annual rate). An interest rate is called nominal if the frequency of (i.e., compounding interest uncorrected for the frequency of compounding).^{[citation needed]}

Loans often include various non-interest charges and fees. One example are points Points, sometimes also called a "discount point", are a form of pre-paid interest. One point equals one percent of the loan amount. By charging a borrower points, a lender effectively increases the yield on the loan above the amount of the stated interest rate. Borrowers can offer to pay a lender points as a method to reduce the interest on a mortgage loan A mortgage loan is a loan secured by real property through the use of a document which evidences the existence of the loan and the encumbrance of that realty through the granting of a mortgage which secures the loan. However, the word mortgage alone, in everyday usage, is most often used to mean mortgage loan in the United States. When such fees are present, lenders are regularly required to provide information on the 'true' cost of finance, often expressed as an annual percentage rate (APR). The APR attempts to express the total cost of a loan as an interest rate after including the additional fees and expenses, although details may vary by jurisdiction.

In economics, continuous compounding is often used due to its particular mathematical properties.^{[citation needed]}

Fixed and floating rates

Commercial loans generally use simple interest, but they may not always have a single interest rate over the life of the loan. Loans for which the interest rate does not change are referred to as fixed rate loans. Loans may also have a changeable rate over the life of the loan based on some reference rate (such as LIBOR and EURIBOR), usually plus (or minus) a fixed margin. These are known as floating rate, variable rate or adjustable rate loans.

Combinations of fixed-rate and floating-rate loans are possible and frequently used. Loans may also have different interest rates applied over the life of the loan, where the changes to the interest rate are governed by specific criteria other than an underlying interest rate. An example would be a loan that uses specific periods of time to dictate specific changes in the rate, such as a rate of 5% in the first year, 6% in the second, and 7% in the third.^{[citation needed]}

Composition of interest rates

In economics, interest is considered the price of credit, therefore, it is also subject to distortions due to inflation. The nominal interest rate, which refers to the price before adjustment to inflation, is the one visible to the consumer (i.e., the interest tagged in a loan contract, credit card statement, etc). Nominal interest is composed of the real interest rate plus inflation, among other factors. A simple formula for the nominal interest is:

i = r + π

Where i is the nominal interest, r is the real interest and π is inflation.

This formula attempts to measure the value of the interest in units of stable purchasing power. However, if this statement were true, it would imply at least two misconceptions. First, that all interest rates within an area that shares the same inflation (that is, the same country) should be the same. Second, that the lenders know the inflation for the period of time that they are going to lend the money.

One reason behind the difference between the interest that yields a treasury bond and the interest that yields a mortgage loan is the risk that the lender takes from lending money to an economic agent. In this particular case, a government is more likely to pay than a private citizen. Therefore, the interest rate charged to a private citizen is larger than the rate charged to the government.

To take into account the information asymmetry aforementioned, both the value of inflation and the real price of money are changed to their expected values resulting in the following equation:

i_t = r_{(t + 1)} + π_{(t + 1)} + σ

Here, i_t is the nominal interest at the time of the loan, r_{(t + 1)} is the real interest expected over the period of the loan, π_{(t + 1)} is the inflation expected over the period of the loan and σ is the representative value for the risk engaged in the operation.

Cumulative interest or return

The calculation for cumulative interest is (FV/PV)-1. It ignores the 'per year' convention and assumes compounding at every payment date. It is usually used to compare two long term opportunities.^{[citation needed]}

Other conventions and uses

Exceptions:

• US and Canadian T-Bills (short term Government debt) have a different calculation for interest. Their interest is calculated as (100-P)/P where 'P' is the price paid. Instead of normalizing it to a year, the interest is prorated by the number of days 't': (365/t)*100. (See also: Day count convention). The total calculation is ((100-P)/P)*((365/t)*100). This is equivalent to calculating the price by a process called discounting at a simple interest rate.
• Corporate Bonds are most frequently payable twice yearly. The amount of interest paid is the simple interest disclosed divided by two (multiplied by the face value of debt).

Flat Rate Loans and the Rule of 78s: Some consumer loans have been structured as flat rate loans, with the loan outstanding determined by allocating the total interest across the term of the loan by using the "Rule of 78s" or "Sum of digits" method. Seventy-eight is the sum of the numbers 1 through 12, inclusive. The practice enabled quick calculations of interest in the pre-computer days. In a loan with interest calculated per the Rule of 78s, the total interest over the life of the loan is calculated as either simple or compound interest and amounts to the same as either of the above methods. Payments remain constant over the life of the loan; however, payments are allocated to interest in progressively smaller amounts. In a one-year loan, in the first month, 12/78 of all interest owed over the life of the loan is due; in the second month, 11/78; progressing to the twelfth month where only 1/78 of all interest is due. The practical effect of the Rule of 78s is to make early pay-offs of term loans more expensive. For a one year loan, approximately 3/4 of all interest due is collected by the sixth month, and pay-off of the principal then will cause the effective interest rate to be much higher than the APY used to calculate the payments. ^[5]

In 1992, the United States outlawed the use of "Rule of 78s" interest in connection with mortgage refinancing and other consumer loans over five years in term.^[6] Certain other jurisdictions have outlawed application of the Rule of 78s in certain types of loans, particularly consumer loans. ^[5]

Rule of 72: The "Rule of 72" is a "quick and dirty" method for finding out how fast money doubles for a given interest rate. For example, if you have an interest rate of 6%, it will take 72/6 or 12 years for your money to double, compounding at 6%. This is an approximation that starts to break down above 10%.

Market interest rates

There are markets for investments (which include the money market, bond market, as well as retail financial institutions like banks) set interest rates. Each specific debt takes into account the following factors in determining its interest rate:

Opportunity cost

This encompasses any other use to which the money could be put, including lending to others, investing elsewhere, holding cash (for safety, for example), and simply spending the funds.

Inflation

Since the lender is deferring his consumption, he will at a bare minimum, want to recover enough to pay the increased cost of goods due to inflation. Because future inflation is unknown, there are three tactics.

• Charge X% interest 'plus inflation'. Many governments issue 'real-return' or 'inflation indexed' bonds. The principal amount or the interest payments are continually increased by the rate of inflation. See the discussion at real interest rate.
• Decide on the 'expected' inflation rate. This still leaves both parties exposed to the risk of 'unexpected' inflation.
• Allow the interest rate to be periodically changed. While a 'fixed interest rate' remains the same throughout the life of the debt, 'variable' or 'floating' rates can be reset. There are derivative products that allow for hedging and swaps between the two.

Default

There is always the risk the borrower will become bankrupt, abscond or otherwise default on the loan. The risk premium attempts to measure the integrity of the borrower, the risk of his enterprise succeeding and the security of any collateral pledged. For example, loans to developing countries have higher risk premiums than those to the US government due to the difference in creditworthiness. An operating line of credit to a business will have a higher rate than a mortgage loan.

The creditworthiness of businesses is measured by bond rating services and individual's credit scores by credit bureaus. The risks of an individual debt may have a large standard deviation of possibilities. The lender may want to cover his maximum risk, but lenders with portfolios of debt can lower the risk premium to cover just the most probable outcome.

Deferred consumption

Charging interest equal only to inflation will leave the lender with the same purchasing power, but he would prefer his own consumption sooner rather than later. There will be an interest premium of the delay. He may not want to consume, but instead would invest in another product. The possible return he could realize in competing investments will determine what interest he charges.

Length of time

Shorter terms have less risk of default and inflation because the near future is easier to predict. Broadly speaking, if interest rates increase, then investment decreases due to the higher cost of borrowing (all else being equal).

Interest rates are generally determined by the market, but government intervention - usually by a central bank- may strongly influence short-term interest rates, and is used as the main tool of monetary policy. The central bank offers to buy or sell money at the desired rate and, due to their control of certain tools (such as, in many countries, the ability to print money) they are able to influence overall market interest rates.

Investment can change rapidly in response to changes in interest rates, affecting national income, and, through Okun's Law, changes in output affect unemployment.{Fact|date=January 2009}

Open market operations in the United States

The Federal Reserve (Fed) implements monetary policy largely by targeting the federal funds rate. This is the rate that banks charge each other for overnight loans of federal funds. Federal funds are the reserves held by banks at the Fed.

Open market operations are one tool within monetary policy implemented by the Federal Reserve to steer short-term interest rates. Using the power to buy and sell treasury securities, the Open Market Desk at the Federal Reserve Bank of New York can supply the market with dollars by purchasing T-notes, hence increasing the nation's money supply. By increasing the money supply or Aggregate Supply of Funding (ASF), interest rates will fall due to the excess of dollars banks will end up with in their reserves. Excess reserves may be lent in the Fed funds market to other banks, thus driving down rates.

Interest rates and credit risk

It is increasingly recognized that the business cycle, interest rates and credit risk are tightly interrelated. The Jarrow-Turnbull model was the first model of credit risk that explicitly had random interest rates at its core. Lando (2004), Darrell Duffie and Singleton (2003), and van Deventer and Imai (2003) discuss interest rates when the issuer of the interest-bearing instrument can default.

Money and inflation

Loans, bonds, and shares have some of the characteristics of money and are included in the broad money supply.

By setting i*n, the government institution can affect the markets to alter the total of loans, bonds and shares issued. Generally speaking, a higher real interest rate reduces the broad money supply.

Through the quantity theory of money, increases in the money supply lead to inflation. This means that interest rates can affect inflation in the future.^{[citation needed]}

Interest in mathematics

It is thought that Jacob Bernoulli discovered the mathematical constant e by studying a question about compound interest.^{[citation needed]}

He realized that if an account that starts with $1.00 and pays 100% interest per year, at the end of the year, the value is $2.00; but if the interest is computed and added twice in the year, the $1 is multiplied by 1.5 twice, yielding $1.00×1.5² = $2.25. Compounding quarterly yields $1.00×1.25⁴ = $2.4414…, and so on.

Bernoulli noticed that this sequence can be modeled as follows:

where n is the number of times the interest is to be compounded in a year.

Formulae

The balance of a loan with regular monthly payments is augmented by the monthly interest charge and decreased by the payment so,

.

where,

i = loan rate/100 = annual rate in decimal form (e.g. 10% = 0.10 The loan rate is the rate used to compute payments and balances.)

r = period rate = i/12 for monthly payments (customary usage for convenience)[2]

B₀ = initial balance (loan principal)

B_k = balance after k payments

k = balance index

p = period (monthly) payment

By repeated substitution one obtains expressions for B_k, which are linearly proportional to B₀ and p and use of the formula for the partial sum of a geometric series results in,

A solution of this expression for p in terms of B₀ and B_n reduces to,

To find the payment if the loan is to be paid off in n payments one sets B_n = 0.

The PMT function found in spreadsheet programs can be used to calculate the monthly payment of a loan:

An interest-only payment on the current balance would be,

The total interest, I_T, paid on the loan is,

The formulas for a regular savings program are similar but the payments are added to the balances instead of being subtracted and the formula for the payment is the negative of the one above. These formulas are only approximate since actual loan balances are affected by rounding. To avoid an underpayment at the end of the loan, the payment must be rounded up to the next cent. The final payment would then be (1+r)B_n-1.

Consider a similar loan but with a new period equal to k periods of the problem above. If r_k and p_k are the new rate and payment, we now have,

Comparing this with the expression for B_k above we note that,

The last equation allows us to define a constant that is the same for both problems,

and B_k can be written,

Solving for r_k we find a formula for r_k involving known quantities and B_k, the balance after k periods,

Since B₀ could be any balance in the loan, the formula works for any two balances separate by k periods and can be used to compute a value for the annual interest rate.

B* is a scale invariant since it does not change with changes in the length of the period.

Rearranging the equation for B^* one gets a transformation coefficient (scale factor),

(see binomial theorem)

and we see that r and p transform in the same manner,

The change in the balance transforms likewise,

which gives an insight into the meaning of some of the coefficients found in the formulas above. The annual rate, r₁₂, assumes only one payment per year and is not an "effective" rate for monthly payments. With monthly payments the monthly interest is paid out of each payment and so should not be compounded and an annual rate of 12·r would make more sense. If one just made interest-only payments the amount paid for the year would be 12·r·B₀.

Substituting p_k = r_k B* into the equation for the B_k we get,

Since B_n = 0 we can solve for B*,

Substituting back into the formula for the B_k shows that they are a linear function of the r_k and therefore the λ_k,

This is the easiest way of estimating the balances if the λ_k are known. Substituting into the first formula for B_k above and solving for λ_k+1 we get,

λ₀ and λ_n can be found using the formula for λ_k above or computing the λ_k recursively from λ₀ = 0 to λ_n.

Since p=rB* the formula for the payment reduces to,

and the average interest rate over the period of the loan is,

which is less than r if n>1.

See also

• Actuarial notation
• Promissory note
• Rate of return
• Cash accumulation equation
• Compound interest
• Credit rating agency
• Credit card interest
• Discount
• Fisher equation
• Hire purchase
• Interest expense
• Interest rate
• Leasing
• Monetary policy
• Mortgage loan
• Risk-free interest rate
• Yield curve
• Time value of money
• Usury
• Simple Interest
• Riba
• JAK members bank a Swedish interest-free bank

References

Specific references

1. ^ Sullivan, arthur; Steven M. Sheffrin (2003). Economics: Principles in action. Upper Saddle River, New Jersey 07458: Pearson Prentice Hall. pp. 261. ISBN 0-13-063085-3. http://www.pearsonschool.com/index.cfm?locator=PSZ3R9&PMDbSiteId=2781&PMDbSolutionId=6724&PMDbCategoryId=&PMDbProgramId=12881&level=4.
2. ^ Sullivan, arthur; Steven M. Sheffrin (2003). Economics: Principles in action. Upper Saddle River, New Jersey 07458: Pearson Prentice Hall. pp. 506. ISBN 0-13-063085-3. http://www.pearsonschool.com/index.cfm?locator=PSZ3R9&PMDbSiteId=2781&PMDbSolutionId=6724&PMDbCategoryId=&PMDbProgramId=12881&level=4.
3. ^ http://scripturetext.com/deuteronomy/23-19.htm
4. ^ http://www.uh.edu/engines/epi2547.htm
5. ^ ^{a b} [1]
6. ^ 15 U.S.C. § 1615

General references

• Duffie, Darrell and Kenneth J. Singleton (2003). Credit Risk: Pricing, Measurement, and Management. Princeton University Press. ISBN 13 978-0691090467.
• Kellison, Stephen G. (1970). The Theory of Interest. Richard D. Irwin, Inc.. Library of Congress Catalog Card No. 79-98251.
• Lando, David (2004). Credit Risk Modeling: Theory and Applications. Princeton University Press. ISBN 13 978-0691089294.
• van Deventer, Donald R. and Kenji Imai (2003). Credit Risk Models and the Basel Accords. John Wiley & Sons. ISBN 13 978-0470820919.

Deepak Tiwari(Rizvi college)BBI student

External links

• Simple Interest Calculator
• Compound Interest Calculator
• Mortgage Payment Calculator
• News and information on mortgages
• White Paper: More than Math, The Lost Art of Interest calculation
• Mortgages made clear Financial Services Authority (UK)
• Simple & Compound Interest Calculator for Multiple Credits and Debits
• OECD interest rate statistics

• You can see a list of current interest rates at these sites:
  • World Interest Rates
  • Forex Motion
  • "Which way to pay"

Categories: Debt | Interest

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• Loan
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• Loans
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